Number Sequence Calculator
Find the nth term and the sum of arithmetic, geometric and Fibonacci sequences, or paste a list of terms to identify the pattern.
aₙ = a₁ + (n − 1)d, adding d each time.
The 10th term of 2, 5, 8, … is 29.
Sum of the first 10 terms
Add every term from a₁ to a₁₀ with Sₙ = n(a₁ + aₙ) ÷ 2.
Check: the first and last terms average (2 + 29) ÷ 2 = 15.5, and 10 × 15.5 = 155.
Your calculation
The same steps the calculator runs, with your numbers.
Terms and running sums
Every term from a₁ to a₁₀.
Values are rounded to 12 significant digits. Very large or very small results use scientific notation.
What Is a Number Sequence?
A number sequence is an ordered list of numbers that follows a rule. Each number is a term, and its position is the index n: in 2, 5, 8, 11 the first term a₁ is 2 and the fourth term a₄ is 11. A sequence can be finite, ending after a set number of terms, or infinite, continuing forever.
A rule can be written two ways. An explicit formula gives any term from its position, such as aₙ = 3n − 1. A recursive formula gives each term from the ones before it, such as aₙ = aₙ₋₁ + 3 with a₁ = 2. Both describe the same sequence.
Three families cover most problems. An arithmetic sequence adds the same amount each time, a geometric sequence multiplies by the same amount, and the Fibonacci sequence adds the two previous terms. Adding the terms of a sequence gives a series, and this calculator finds that sum as well.
How to Use the Number Sequence Calculator
- Pick the sequence type. Choose Arithmetic if the same number is added each time, Geometric if each term is multiplied by the same number, Fibonacci for 1, 1, 2, 3, 5 and so on, or Find pattern if you have a list and do not know the rule.
- Enter the values. For arithmetic, enter the first term and the common difference d. For geometric, enter the first term and the common ratio r. Then enter n, the position of the term you want. For Find pattern, paste 3 to 30 terms separated by commas.
- Read the answer. The large number is the nth term, or the next term in pattern mode. The card below gives the sum of the first n terms, or the formula for the nth term. Then come the steps with your numbers and a table of terms.
Results update as you type. The quick examples cover an arithmetic sequence, the odd numbers, doubling, halving, the 30th Fibonacci number and the square numbers.
Arithmetic Sequence Formula
The difference d is any term minus the term before it. The sum formula multiplies the number of terms by the average of the first and last terms. For 1 to 100: 100 × (1 + 100) ÷ 2 = 5,050.
Geometric Sequence Formula
The ratio r is any term divided by the term before it. When r = 1 every term is equal and Sₙ = n × a₁.
Fibonacci Sequence Formula
The Fibonacci sequence starts 1, 1 and adds the two previous terms to get the next: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. The recursive rule is Fₙ = Fₙ₋₁ + Fₙ₋₂ with F₁ = F₂ = 1, which matches F₀ = 0.
The closed form is called Binet's formula. The ratio of consecutive Fibonacci numbers approaches the golden ratio φ: F₃₁ ÷ F₃₀ = 1,346,269 ÷ 832,040 ≈ 1.6180339888. The calculator uses exact whole number arithmetic, so even F₁₀₀ = 354,224,848,179,261,915,075 is shown digit for digit.
How to Find the Pattern in a Sequence Manually
To find the next number in a sequence, test the simplest rules first. The pattern finder runs the same checks in this order.
- First differences. Subtract each term from the next. If every difference is equal, the sequence is arithmetic.
- Ratios. Divide each term by the one before. If every ratio is equal, the sequence is geometric.
- Sum of two terms. If each term equals the two before it added together, it follows the Fibonacci rule.
- Second differences. Take the differences of the differences. A constant second difference means a quadratic formula aₙ = An² + Bn + C. A constant third difference means a cubic formula.
Worked example: 3, 7, 13, 21, 31
The first differences are 4, 6, 8, 10, which change, so the sequence is not arithmetic. The ratios 7 ÷ 3 and 13 ÷ 7 differ, so it is not geometric. The second differences are 2, 2, 2, so the formula is quadratic.
Common Number Sequences Chart
These sequences come up often in school problems and puzzles. Paste any of them into Find pattern to see the formula and the next terms.
| Sequence | First terms | Type | nth term |
|---|---|---|---|
| Natural numbers | 1, 2, 3, 4, 5 | Arithmetic, d = 1 | n |
| Even numbers | 2, 4, 6, 8, 10 | Arithmetic, d = 2 | 2n |
| Odd numbers | 1, 3, 5, 7, 9 | Arithmetic, d = 2 | 2n − 1 |
| Multiples of 5 | 5, 10, 15, 20, 25 | Arithmetic, d = 5 | 5n |
| Square numbers | 1, 4, 9, 16, 25 | Quadratic | n² |
| Triangular numbers | 1, 3, 6, 10, 15 | Quadratic | n(n + 1) ÷ 2 |
| Cube numbers | 1, 8, 27, 64, 125 | Cubic | n³ |
| Powers of 2 | 2, 4, 8, 16, 32 | Geometric, r = 2 | 2ⁿ |
| Powers of 10 | 10, 100, 1,000, 10,000 | Geometric, r = 10 | 10ⁿ |
| Halving from 1 | 1, 0.5, 0.25, 0.125 | Geometric, r = 0.5 | 0.5ⁿ⁻¹ |
| Fibonacci | 1, 1, 2, 3, 5, 8 | Recursive | Fₙ₋₁ + Fₙ₋₂ |
| Prime numbers | 2, 3, 5, 7, 11 | No simple rule | None |
Arithmetic vs Geometric Sequences
| Arithmetic | Geometric | |
|---|---|---|
| Rule | Add d | Multiply by r |
| nth term | a₁ + (n − 1)d | a₁ × rⁿ⁻¹ |
| Sum of n terms | n(a₁ + aₙ) ÷ 2 | a₁(1 − rⁿ) ÷ (1 − r) |
| Infinite sum | Only if every term is 0 | a₁ ÷ (1 − r) if −1 < r < 1 |
| Graph | Straight line | Exponential curve |
| Example | 5, 9, 13, 17 | 5, 15, 45, 135 |
For 5, 9, 13 the 25th term is 101 and the first 25 terms add to 1,325. For 5, 15, 45 the 6th term is 1,215 and the first 6 terms add to 1,820.
Understanding Your Result
The nth term is a single number in the sequence. The sum Sₙ adds every term from the first through the nth, so it grows much faster. For 2, 5, 8 the 10th term is 29 but the sum of the first 10 terms is 155.
A geometric sum with a ratio between −1 and 1 levels off toward the infinite sum: 100, 50, 25 reaches 199.21875 after 8 terms and 199.8046875 after 10, getting closer to 200. With a ratio above 1, the last term makes up most of the sum.
In pattern mode, the calculator reports the simplest rule that fits. A short list can fit more than one rule, so enter four or more terms when you can.
Where Number Sequences Are Used
Seats in a theater (arithmetic)
A theater has 20 seats in the first row and each row has 2 more than the row in front. Row 15 has 20 + 14 × 2 = 48 seats, and the 15 rows hold 15 × (20 + 48) ÷ 2 = 510 seats.
Compound interest (geometric)
A balance of $1,000 earning 5% a year is multiplied by 1.05 each year, a geometric sequence with r = 1.05. After 10 years it is 1,000 × 1.05¹⁰ = $1,628.89. Because the starting balance is term 1, the balance after 10 years is term 11. The Compound Interest Calculator handles deposits and different compounding periods.
A bouncing ball (geometric)
A ball dropped from 10 m that rebounds to 60% of its previous height reaches 6 m, then 3.6 m, then 2.16 m. The 5th rebound is 6 × 0.6⁴ = 0.7776 m, and all the rebound heights add up to 6 ÷ (1 − 0.6) = 15 m.
Common Mistakes
- Using n instead of n − 1. The first term already counts as one step, so the 10th term of 2, 5, 8 is 2 + 9 × 3 = 29, not 2 + 10 × 3 = 32.
- Confusing the term with the sum. The 10th term of 2, 5, 8 is 29. The sum of the first 10 terms is 155.
- Subtracting in the wrong order. The difference is the later term minus the earlier one. For 20, 17, 14 the difference is −3, not 3.
- Using the infinite sum when |r| ≥ 1. For doubling, the terms keep growing, so no infinite sum exists even though a₁ ÷ (1 − r) gives a number.
- Mixing Fibonacci starting points. This page uses F₁ = F₂ = 1. Sources that call 0 the first Fibonacci number shift every position by one.
- Trusting three terms. 2, 4, 8 could be doubling or a quadratic rule that gives 14 next. More terms make the pattern clear.
Related Calculations
Geometric sequences rely on powers, which the Exponent Calculator evaluates for any base. To find how many terms a geometric sequence needs to reach a target, solve rⁿ⁻¹ = target ÷ a₁ with the Log Calculator: doubling from 1 passes 1,000,000 at term 21, because 2²⁰ = 1,048,576.
The average of an arithmetic sequence equals the average of its first and last terms, and the Mean, Median, Mode Calculator finds the average of any list. The LCM Calculator finds where two arithmetic sequences of multiples first meet.
Frequently Asked Questions
How do you find the nth term of an arithmetic sequence?
Use aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference, the amount added each time. For 2, 5, 8, 11, the first term is 2 and the difference is 3, so the 10th term is 2 + (10 − 1) × 3 = 2 + 27 = 29. You can also simplify the formula to aₙ = 3n − 1 and substitute any n directly. To find d from a list, subtract any term from the one after it: 5 − 2 = 3. If the differences are not all equal, the sequence is not arithmetic.
What is the formula for the sum of a geometric sequence?
The sum of the first n terms is Sₙ = a₁(1 − rⁿ) ÷ (1 − r), where a₁ is the first term and r is the common ratio, with r not equal to 1. For 3, 6, 12, 24, the ratio is 2, so the sum of the first 8 terms is 3 × (1 − 2⁸) ÷ (1 − 2) = 3 × (−255) ÷ (−1) = 765. When r equals 1, every term is the same and the sum is n × a₁. When the ratio is between −1 and 1, the infinite sum exists and equals a₁ ÷ (1 − r), so 100, 50, 25 and so on add up to 200.
What is the 10th Fibonacci number?
The 10th Fibonacci number is 55, counting the sequence as 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 with F₁ = 1 and F₂ = 1. Each number is the sum of the two before it, so F₁₁ = 34 + 55 = 89 and F₁₂ = 55 + 89 = 144. Some sources list 0 as the first number, which moves every label by one. The sum of the first 10 Fibonacci numbers is 143, which equals F₁₂ − 1. The 30th Fibonacci number is 832,040, and the ratio of consecutive terms approaches the golden ratio, about 1.618034.
How can you tell if a sequence is arithmetic or geometric?
Subtract each term from the next one. If every difference is the same, the sequence is arithmetic: 5, 9, 13, 17 has a constant difference of 4. If the differences change, divide each term by the one before it. If every ratio is the same, the sequence is geometric: 5, 15, 45, 135 has a constant ratio of 3. If neither works, check the differences of the differences. A constant second difference means a quadratic sequence, such as 3, 7, 13, 21, 31, whose formula is n² + n + 1. A sequence where each term is the sum of the two before it follows the Fibonacci rule.
What is the sum of the numbers from 1 to 100?
The sum of the whole numbers from 1 to 100 is 5,050. The numbers form an arithmetic sequence with first term 1, last term 100 and 100 terms, so Sₙ = n(a₁ + aₙ) ÷ 2 = 100 × (1 + 100) ÷ 2 = 5,050. Pairing works too: 1 + 100, 2 + 99, 3 + 98 and so on gives 50 pairs that each add to 101, and 50 × 101 = 5,050. The same formula gives the sum of the first 50 odd numbers, 1 + 3 + 5 through 99, which is 50 × (1 + 99) ÷ 2 = 2,500.
Sources & References
- OpenStax. College Algebra 2e, 9.1 Sequences and Their Notations (explicit and recursive formulas). openstax.org
- OpenStax. College Algebra 2e, 9.2 Arithmetic Sequences. openstax.org
- OpenStax. College Algebra 2e, 9.3 Geometric Sequences. openstax.org
- OpenStax. College Algebra 2e, 9.4 Series and Their Notations (arithmetic, geometric and infinite sums). openstax.org
- NIST Digital Library of Mathematical Functions. 24.15 Related Sequences of Numbers, Fibonacci and Lucas Numbers. dlmf.nist.gov
- The OEIS Foundation. A000045 Fibonacci numbers. oeis.org
Results are for educational purposes. Every worked number on this page was recomputed with the same formulas the calculator uses. Methodology reviewed: 2026.